how to find the arc length

What Is Arc Length?

The arc length is the measure of the distance along the circumference of a circle making up the arc. To find the measure of the arc length of a circle, remember you are really finding the fractional amount of the whole circle represented by that arc. Knowing the measure of the central angle in either degrees or radians and the entire circumference, you can find the exact length of the arc.

1. Arc Length Of A Circle
2. Circumference and Arc Length
3. Arc Length Formulas
• Degrees
• Radians
4. How To Find Arc Length
5. Arc Length Examples

What You Need To Know

A circle has a measurable distance around its outside, the circumference. Circles also have diameters, line segments from the circle, through the center, to the circle on the other side. Circles have radii or radiuses, which are half diameters (from the circle to the center). Any portion of a circumference, such as would be created by two radii some known central angle apart, is an arc.

Arc Length Of A Circle

Angles in circles can be measured in degrees ($360°$) or radians. Radians are the metric measure for angles. One radian is the angle formed when an arc of a circle is equal in length to the radius of that circle. That means any circle has a circumference of $2\pi$ radians. One radian is roughly equivalent to $57.296°$, though such a conversion is not usually helpful.

Complete all your work in either degrees or radians. Try to avoid leaping between units.

Degree arc measures of circles are notated by the italic letter $m$ (for measure) followed by the two endpoints of the arc on the circle, with a tiny arc drawn over the two capital letters. So in the circle below, arc $\stackrel{⌢}{AB}$ has an angle measure of $36°$. The notation would be $m\stackrel{⌢}{AB}$.

Circumference and Arc Length

A circle has a special numerical relationship between circumference and either diameter or radius. If you know the radius, $r$, you can use that to find the circumference, $C$, using the formula $C=2\pi r$ . If you know the diameter, $d$, then:

$C=\pi d$

We use $3.14159$ as an approximation of the value of $\pi$, which you probably remember is a non-repeating, non-terminating number. Calculators with $\pi$ keys are far more accurate, but a decimal expressed to 100,000ths is very accurate, too.

Arc Length Formulas

You can use a formula for either degrees or radians for finding arc length. Let's go over each arc length equation step-by-step.

Arc Length Formula For Degrees

The length of an arc is found by forming a ratio of the arc to the whole circle, then multiplying it by the formula for circumference, either $2\pi r$ or $\pi d$, like this:

$Arclengt{h}_{\stackrel{⌢}{AB}}=\left(\frac{m\stackrel{⌢}{AB}}{360°}\right)\left(2\pi r\right)$

Or, if you have the diameter:

$Arclengt{h}_{\stackrel{⌢}{AB}}=\left(\frac{m\stackrel{⌢}{AB}}{360°}\right)\left(\pi d\right)$

Arc Length Formula For Radians

When working with radians, the formula is even simpler, where $\theta$ (Theta) is the central angle in radians and $r$ is the radius:

How To Find Arc Length

Finding Arc Length Using Degrees

Take another look at the circle above, with $\angle ACB$ measured as $36°$ and radii of $30cm$. If you put that angle ($36°$) and that radius ($30cm$) into the arc length formula used for degrees, you get this:

Start with our formula:

$Arclengt{h}_{\stackrel{⌢}{AB}}=\left(\frac{m\stackrel{⌢}{AB}}{360°}\right)\left(2\pi r\right)$

Plug in what we know:

$=\left(\frac{36°}{360°}\right)\left(2·\pi ·30\right)$

The fraction is $\frac{1}{10}th$ the circumference. Multiply $2\pi r$ times $\frac{1}{10}$, using $3.14159$ as a very close approximation of $\pi$ if you do not have a calculator with a $\pi$ key:

$=\frac{1}{10}·2·\pi ·30$

$=\frac{1}{10}·60\pi$

$=6\pi$

$\mathbf{\approx }\mathbf{}\mathbf{18.84954}\mathbf{}\mathbf{cm}$

Because $36°$ is $\frac{1}{10}$ of $360°$, the length of arc $\stackrel{⌢}{AB}$ is $\frac{1}{10}$ of the circle's circumference.

Finding Arc Length Using Radians

Using the radians formula is even easier. For our same circle, the angle in radians is $0.628319rad$, so we use that instead of degrees:

Start with our formula:

$Arclength=\theta r$

$=\theta ·30$

Let's convert Theta to a number we can use:

$=0.628319·30$

$\mathbf{=}\mathbf{18.84957}\mathbf{cm}$

Note the two measurements: they differ by only $0.00003cm$ due to rounding errors. A calculator with $\pi$ would be even more precise.

Let's Check Our Work

We can test both these measurements, because we set the central angle, $\angle ACB$ to be a convenient fraction of our circle. We made it $36°$, or $\frac{1}{10}th$. That means we can check the fractional amount of the circumference by using $2\pi r$ to find the circumference and then take $\frac{1}{10}th$ of that:

$Circumference=2·\pi ·r$

$Circumference=2·\pi ·30$

$Circumference=2·3.14159·30$

$Circumference=188.4954cm$

$\frac{1}{10}$ of the Circumference = $18.84954cm$

It checks out!

Arc Length Examples

Here are two more arc length problems to try, one with a central angle measured in degrees and one measured in radians. Remember, you can only calculate the arc length if you know that central angle.

Arc Length Problem #1

[insert drawing of Circle A with points W and Y on the circle creating radii AW and AY and a central angle of 22.5° {1/16th of the circle} and labeled radius of 5 yards]

Take a moment to jot down your important facts, Then step away and do your calculations. Once you have your answers, see if your work matches this work.

$Arclengt{h}_{\stackrel{⌢}{WY}}=\left(\frac{m\stackrel{⌢}{WY}}{360°}\right)\left(2\pi r\right)$

$=\left(\frac{22.5°}{360°}\right)\left(2·\pi ·5\right)$

$=\frac{1}{16}·10\pi$

- or -

$=0.0625·10\pi$

You can express the fractional part of the circle either as a fraction or decimal. In either case, you get the same answer:

$\mathbf{=}\mathbf{}\mathbf{1.96349}\mathbf{yards}$

Arc Length Problem #2

[insert drawing of Circle E with points Y and 2 on the circle creating radii EY and ES and a central angle labeled 1.0472 rad {60° angle, 1/6th of the circle} and labeled radius of 3 meters]

We'll need to use the formula for radians:

$Arclength=\theta r$

$=1.0472rad·3$

$\mathbf{=}\mathbf{}\mathbf{3.1416}\mathbf{meters}$

Next Lesson:

How To Find Arc Measure Formula

how to find the arc length

Source: https://tutors.com/math-tutors/geometry-help/how-to-find-arc-length-formula

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